Answer :
Sure! Let's factor the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] completely.
1. Identify and factor out the greatest common factor (GCF):
The expression [tex]\( x^3 - 7x^2 + 10x \)[/tex] has a common factor in all terms, which is [tex]\( x \)[/tex]. Therefore, we factor [tex]\( x \)[/tex] out of each term:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the quadratic [tex]\( x^2 - 7x + 10 \)[/tex]:
Now, focus on the quadratic expression [tex]\( x^2 - 7x + 10 \)[/tex]. We need to factor this into two binomials. We are looking for two numbers that multiply to 10 and add to -7.
After examining the factors of 10, we find that -5 and -2 work because:
- Multiplying: [tex]\((-5) \times (-2) = 10\)[/tex]
- Adding: [tex]\((-5) + (-2) = -7\)[/tex]
Thus, the quadratic [tex]\( x^2 - 7x + 10 \)[/tex] can be factored as:
[tex]\[
(x - 5)(x - 2)
\][/tex]
3. Write the completely factored form of the original polynomial:
Combining the factored [tex]\( x \)[/tex] from step 1 with the factored quadratic from step 2, the completely factored form of the expression is:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
Therefore, the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is factored completely as [tex]\( x(x - 5)(x - 2) \)[/tex].
1. Identify and factor out the greatest common factor (GCF):
The expression [tex]\( x^3 - 7x^2 + 10x \)[/tex] has a common factor in all terms, which is [tex]\( x \)[/tex]. Therefore, we factor [tex]\( x \)[/tex] out of each term:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the quadratic [tex]\( x^2 - 7x + 10 \)[/tex]:
Now, focus on the quadratic expression [tex]\( x^2 - 7x + 10 \)[/tex]. We need to factor this into two binomials. We are looking for two numbers that multiply to 10 and add to -7.
After examining the factors of 10, we find that -5 and -2 work because:
- Multiplying: [tex]\((-5) \times (-2) = 10\)[/tex]
- Adding: [tex]\((-5) + (-2) = -7\)[/tex]
Thus, the quadratic [tex]\( x^2 - 7x + 10 \)[/tex] can be factored as:
[tex]\[
(x - 5)(x - 2)
\][/tex]
3. Write the completely factored form of the original polynomial:
Combining the factored [tex]\( x \)[/tex] from step 1 with the factored quadratic from step 2, the completely factored form of the expression is:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
Therefore, the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is factored completely as [tex]\( x(x - 5)(x - 2) \)[/tex].
